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A question for finite 2-group junkies ...

The dihedral group $D_16$ of order 16 has a presentation $D_16= langle a,t | a^2=t^8=atat=erangle$.

Question: Does there exist a finite 2-group $G$ containing $D_16$ as a subgroup, and an element $g in G$ such that $gag^-1=t^4$? Bonus pats-on-the-back if $G$ has order 64.

An obvious reduction: one can assume that $G =langle D_16,grangle$.

An obvious constraint: $D_16$ cannot be normal in $G$ (so $G$ can't have order 32).

[This question has come up in investigations of the Balmer spectrum of $G$--equivariant stable homotopy for finite $p$--groups $G$. Like Dr. Frankenstein, I am looking for interesting subjects to experiment on, and my student Chris Lloyd is serving as the able assistant to the mad scientist.]

No. We can prove this by induction. Let $G$ be the smallest $2$-group in which this situation occurs. Then $G$ has a normal central subgroup $N$ of order $2$.

If $N$ has trivial intersection with the subgroup $langle a,t rangle = D_16$, then the same situation occurs in $G/N$, contradicting the minimality of $G$.

So that intersection must be nontrivial, and hence $N le langle a,t rangle$, and then we must have $N = Z(langle a,t rangle) = langle t^4 rangle$.

But then $t^4 in Z(G)$, contradicting the assumption that it is conjugate in $G$ to $a$.

The situation you describe can occur in finite groups, such as in simple groups $rm PSL(2,q)$ for some prime powers $q$, such as $q=17$, but not in finite $2$-groups.